LECTURES on DEFORMATIONS of G-STRUCTURES the Author Is Thankful to Departamento De Matemáticas, Facultad De Ciencias De

Home , Atlas (topology), Pseudogroup, Symplectic manifold

LECTURES ON DEFORMATIONS OF G-STRUCTURES

M.A.MALAKHALTSEV

The author is thankful to Departamento de Matemáticas, Facultad de Ciencias de Uni- versidad de los Andes (Bogotá, Colombia), and Grupo Sanford for invitation, hospitality, and financial support during his visit to Universidad de los Andes in September-October, 2006. Also he expresses his deep gratitude to professor Jose Ricardo Arteaga Bejarano for useful cooperation and constant support.

1. Pseudogroups. Sheaves

1.1. Pseudogroups. Recall that a transformation group G is a subgroup in a group Diff(M), so each g ∈ G is a diffeomorphism of M. A local diffeomorphism is a diffeomorphism f : U → V , where U and V are open subsets in M. We set D(f) = U, E(f) = V . Certainly, the set of local diffeomorphisms is not a group of transformations because the composition is not well-defined. A generalization of the notion of transformation group is a pseudogroup.

Deﬁnition 1. A pseudogroup Γ of transformations of a smooth manifold M is a collection of local diﬀeomorphisms such that 0 • For any f ∈ Γ and open U ⊂ D(f), we have f|U 0 ∈ Γ;

• Let U = ∪Ui, where Ui are open. Let local diﬀeomorphism f : U → V be such that f| ∈ Γ, then f ∈ Γ. Ui • For any f ∈ Γ, we have f −1 ∈ Γ. • For f, g ∈ Γ such that E(f) ⊂ D(g) we have g ◦ f ∈ Γ;

Example 1.1. All local diﬀeomorphisms of a manifold.

Example 1.2. Pseudogroup of automorphisms of a tensor field: a) pseudogroup of analyt- ical diffeomorphisms b) pseudogroup of symplectic diffeomorphisms.

Example 1.3. The pseudogroup Γ(G) generated by a group G of transformations of a manifold M. The elements of Γ(G) are the restrictions of elements of G to open sets in M. This is a preliminary version of lectures the author gave in the University “Los Andes” in September- October, 2006. 1 2 M.A.MALAKHALTSEV

We can say informally that a subset Γ of local diffeomorphism of M is a pseudogroup if and only if (i) the composition and inverse elements are well-defined, (ii) the elements of Γ are defined by a local condition (in fact, by a condition on germs).

1.2. Sheaves. Here we give only main deﬁnitions and results of the sheaf theory. For a detailed exposition see [11], [12].

Deﬁnition 2. Let M be a topological space. Suppose that to each open U we assign a U group F(U) and if V ⊂ U we have group homomorphism rV : F(U) → F(V ) such that

U V U (1) rV rW = rW U (2) rU = Id

Then we get a presheaf F of groups.

In the same way we can deﬁne presheaves of rings, vector spaces, etc. We will call the elements of A(U) the sections of A over U.

Example 1.4. • The presheaf of smooth sections of a smooth bundle ξ: to each open set U we put U in correspondence the set of sections of ξ over U, and the map rV is the restriction map. • The presheaf of constant functions on a manifold: to each open set U we put in U correspondence the set of constant functions on U, and the map rV is again the restriction map.

Let A, B be preasheaves. A presheaf morphism φ : A → B is the collection of homo-

morphisms φU : A(U) → B(U) given for each open U such that, for every V ⊂ U, we U U have rV φU = φV rV . If A, B are preasheaves of abelian groups, then we can deﬁne the kernel and the image of φ: the kernel of φ is the presheaf given by

U → Ker(φ : A(U) → B(U)),

and the image of φ is the presheaf given by

U → Im(φ : A(U) → B(U)).

Deﬁnition 3. A presheaf A is called a sheaf if

U U (1) If U = ∪Ui and s, t ∈ A(U) and rUi (s) = rUi (t), then s = t; LECTURES ON DEFORMATIONS OF G-STRUCTURES 3

Ui Uj (2) Let U = ∪Ui. Then for any collection si ∈ A(Ui) such that rUi∩Uj (si) = rUi∩Uj (sj), U there exists a (unique) s ∈ A(U) such that rUi (s) = si.

Example 1.5. (1) The presheaf of local sections of a bundle is a sheaf. (2) The presheaf of constant functions on a manifold is not a sheaf.

1.2.1. Passing from a presheaf to a sheaf. Let A be a presheaf on a manifold M. Then, for each x ∈ M, we can deﬁne a germ of a section s ∈ A(U) in the standard way. Namely, we consider the set of pairs (W, s ∈ A(W )) such that x ∈ W and take the equivalence relation on this set deﬁned as follows: (U, s ∈ A(U)) ∼ (V, t ∈ A(V )) if and only if there U V exists x 3 W ⊂ U ∩V such that rW (s) = rW (t). Denote by Ax the set of germs of sections of A at x.

Now we consider the set E(A) = ∪x∈M Ax and introduce a topology on E(A) in the following way. The idea is to set up a minimal topology such that the sections of the presheaf are continuous maps. The prebase of the topology on E(A) consists of sets

(3) Ω = {< s >x| s ∈ A(U)}

Note that this topology has a lot of open sets, and, in general, is not Hausdorﬀ.

We have the natural projection π : E(A) → M, < s >x→ x and thus we obtain a covering (E(A), π, M), in the sense that for each point y ∈ E(A) there exists a neighborhood in E(A) homeomorphic to a neighborhood of x = π(y). Then the sheaf A associated with A is the sheaf of continuous sections of π : E(A) → M. We can describe the sections of b this sheaf in the following way: a section s : U → E(A), x →< σx >x is continuous if for each x ∈ U there exist a neighborhood x 3 W ⊂ U and a section σ ∈ A(U) such that

< sx >x=< σ >x.

Example 1.6. The sheaf associated with the presheaf of constant functions is the sheaf of locally constant functions.

Let A and B be sheaves of abelian groups. If φ : A → B is a sheaf morphism, then we have the image presheaf U → φ(A(U)) which is not a sheaf in general. However we denote by Imφ the sheaf associated with the image sheaf. Also, for sheaves B ⊂ A we deﬁne the presheaf U → A(U)/B(U), and we denote by A/B the sheaf associated to this presheaf.

1.2.2. Sheaf cohomology. Let G be a sheaf of abelian groups on a manifold M. Take at m most countable open covering U = {Ui} of M. Deﬁne the nerve of U be N(U) = tN (U), 4 M.A.MALAKHALTSEV

where

m (4) N (U) = {{i0 . . . im} | Ui0 ∩ Ui2 ∩ . . . Uim =6 ∅}

Now an m-cochain on M is a family of sections {ci0...im ∈ G(Ui0 ∩ Ui2 ∩ . . . Uim ) | m {i0 . . . im} ∈ N (U)}. The order of indices is unessential, so we assume that ci0i1...im is

skew-symmetric with respect to i0, i1, . . . , im. The set Cˇm(U; G) of m-cochains is evidently an abelian group. We have the group homomorphism:

(5) d : Cˇm(U; G) → Cˇm+1(U; G)

m+1 U c k i0...ik...im+1 (dc)i0...im+1 = (−1) r (c b ) Ui0...im+1 i0...ik...im+1 k X=0 For example,

0 1 (6) d : C (U; G) → C (U; G), dcβα = cβ − cα

1 2 (7) d : C (U; G) → C (U; G), dcαβγ = cβγ − cαγ + cαβ

One can prove that d ◦ d = 0 by direct calculations, hence we obtain a complex (Cˇ∗(U; G), d) called the Cech complex of the covering U with coeﬃcients in the sheaf G. Certainly, the cohomology of this complex depends on the covering.

Example 1.7. Hˇ 0(U; G) = G(M).

Now the set of open coverings is a partially ordered set with respect to the inscribing relation ≤, and, if V ≤ U, we have the natural map Hˇ (U; G) → Hˇ (V; G). Thus we can deﬁne the Cech cohomology of M with coeﬃcients in G:

(8) Hˇ (M; G) = lim Hˇ (U; G) −→ The following theorems make it possible to calculate the Cech cohomology without taking the direct limit. Let G be a sheaf of abelian groups on M. An open covering is said to be ‘ﬁne’ if q Hˇ (UI ; G) ∼= 0, q > 0, for each I ∈ N(U).

Theorem 1 (Leray). If U is a ﬁne covering on M, then Hˇ (M; G) ∼= Hˇ (U; G).

An exact sequence of sheaves

i d d (9) 0 → G →− F0 →− F1 →− . . . is called a resolution of G. If all the sheaves F are ﬁne, then we say that (9) is a ﬁne

resolution of G. From the ﬁne resolution 9 we get the complex (Fk(M), d). LECTURES ON DEFORMATIONS OF G-STRUCTURES 5

Theorem 2 (Abstract de Rham Theorem). Given a ﬁne resolution Fk of a sheaf G, we have Hˇ (M; G) ∼= H(F∗(M), d).

Example 1.8. The sheaf RM of locally constant functions on a manifold M admits the fine resolution R 0 d 1 d d n 0 → M → ΩM →− ΩM →− . . . →− ΩM → 0, q R where ΩM is the sheaf of differential q-forms on M, therefore the cohomology of H(M; M ) with coefficients in the sheaf of locally constant functions is isomorphic to the de Rham cohomology.

i p Theorem 3. If 0 → A →− B →− C → 0 is a exact sequence of sheaves, we have a long exact cohomology sequence (10) δ i∗ p∗ δ · · · → Hˇ m−1(M; C) →− Hˇ m(M; A) −→ Hˇ m(M; B) −→ Hˇ m(M; C) →− Hˇ m+1(M; A) → . . .

2. Deformations of pseudogroup structures associated with integrable first order G-structures

We will apply deformation theory to the integrable ﬁrst order G-structures (for the general theory of G-structures we refer the reader to e. g. [14], [15], [13]).

2.1. Integrable G-structures. Let M be a smooth n-dimensional manifold, L(M) → M the frame bundle of M, and X(M) the Lie algebra of vector ﬁelds on M. For a Lie subgroup

G ⊂ GL(n), consider the bundle EG(M) = L(M)/G → M.

Remark 2.1. EG is a natural bundle: this means that EG is the functor Man → Bundles, where Man is the category of smooth manifolds whose morphisms are diffeomorphisms, and Bundles is the category of fiber bundles. This functor sends a manifold M to the 0 fiber bundle EG(M), and a diffeomorphism f : M → M to the natural bundle morphism c 0 f : EG(M) → EG(M ).

Deﬁnition 4. A ﬁrst order G-structure is a section s: M → EG(M).

Example 2.1 (C). An almost complex structure is a section s: M → EG(M), where G =

GL(n, C) ⊂ GL(2n, R). One can easily prove that the sections of EG(M) are in 1-1- correspondence with the linear operator ﬁelds J such that J 2 = −I.

Example 2.2 (F). A distribution of codimension q can be identiﬁed with a section s: M → A 0 EG(M), where G = { } ⊂ GL(n, R), where A ∈ GL(q, q), B ∈ Mat(n − q, q), B C ! C ∈ GL(n − q, n − q). 6 M.A.MALAKHALTSEV

Example 2.3 (S). An almost symplectic structure is a section of s: M → EG(M), where

G = Sp(n) ⊂ GL(2n, R). One can easily prove that the sections of EG(M) are in 1-1- correspondence with non-degenerate diﬀerential 2-forms on M.

Definition 5. We say that a first order G-structure s is integrable iff s is locally equivalent n to a given first order G-structure s0 on R .

Example 2.4 (C). A complex structure is a section s: M → EG(M) locally equivalent to 1 Rn J ∈ T1 ( ) such that J∂k = ∂n+k, J∂n+k = −∂k, k = 1, n.

Example 2.5 (F). A foliation of codimension q is a section s: M → EG(M) locally equivalent to the distribution ∆ given by equations dx1 = 0, dx2 = 0, . . . , dxq = 0.

Example 2.6 (S). A symplectic structure is a section s: M → EG(M) which is locally 1 2 2n−1 2n equivalent to ω0 = dx ∧ dx + · · · + dx ∧ dx .

Remark 2.2. For all the classical structures, like the complex structure, the foliation structure, etc., the section s0 is invariant under the translation group, so s0 is determined by its value at a point 0 ∈ Rn.

2.2. Γ-structure associated with integrable G-structure. Let Γ0 be a pseudogroup n of transformations of R . A Γ0-structure on a manifold M is a maximal atlas {(Uα, φα)} −1 on M whose transition functions φβφα lie in Γ0. n Given an integrable G-structure s : M → L(M)/G, and so the model section s0 : R → n L(R )/G, we obtain the pseudogroup Γ0 = Γ(s0) of automorphisms of s0. For classical G-structures, this pseudogroup is transitive (see the remark above). It is clear that a manifold M is endowed with an integrable G-structure if and only if

M is endowed with Γ(s0)-structure.

For a manifold M endowed with a Γ0-structure {(Uα, φα)}, we have the pseudogroup −1 Γ of diﬀeomorphisms of f : M → M such that the local representations φβfφα lie in Γ0.

In case the Γ0-structure corresponds to an integrable G-structure s : M → L(M)/G, the pseudogroup Γ consists of local automorphims of s: f ∈ Γ if and only if f ∗(s) = s.

A vector field X is called a Γ-vector field if the flow φt consists of elements of Γ. On each manifold M endowed with a Γ0-structure we have the sheaf XΓ of Γ-vector fields.

In case the Γ0-structure corresponds to an integrable G-structure s : M → L(M)/G, a vector ﬁeld X lies in XΓ(U) if and only if LX s = 0. Now we continue considering the above examples and present the corresponding pseudogroups Γ0, as well as the sheaves of Γ0-vector ﬁelds. LECTURES ON DEFORMATIONS OF G-STRUCTURES 7

2n 2n Example 2.7 (C). A complex structure. Γ0 consists of local diﬀeomorphisms f : R → R

which satisfy the Cauchy-Riemann equations, i.e. df commutes with J0, where

0 −I J0 = . I 0 !

The sheaf XΓ of Γ-vector ﬁelds on M consists of holomorphic vectors ﬁelds of (M, J).

Example 2.8 (F). A foliation of codimension q. Let (xa, xα), a = 1, q, α = q + 1, n, be n n n coordinates on R . The pseudogroup Γ0 consists of local diffeomorphisms f : R → R whose germs are of the form ya = f a(xb), yα = f α(xb, xβ). In fact, a local diffeomorphism ∂f a lies in Γ if and only if ∂xα = 0. And the sheaf XΓ consists of vector fields V with local expression a b α b β V = V (x )∂a + V (x , x )∂α

Example 2.9 (S). A symplectic structure. A local diﬀeomorphism lies in Γ0 if and only if ∗ f ω0 = ω0. The sheaf XΓ consists of vector ﬁelds V which are locally Hamiltonian vector

ﬁelds: V ∈ X(U) if and only if LV ω = 0 if and only if diV ω = 0 if and only if V has local k ks expression V = ω ∂kf.

2.3. Deformations of pseudogroup structure. Let us recall deﬁnitions and results of the theory of deformations of pseudogroup structures (see [5]–[7] and also [8], [9]). Let Γ be a pseudogroup of local diﬀeomorphisms of a manifold M.

Deﬁnition 6. A one-parameter family φs of elements of Γ is said to be smooth if U =

{(x, s) ∈ M × R | x ∈ D(φs)}, where D(g) is the domain of g ∈ Γ, is open in M × R and n Φ : U → R , Φ(x, s) = φs(x), is smooth. A smooth one-parameter family of elements of Γ will be called a curve in Γ.

Let φs, |s| < ε, be a curve in Γ such that φ0 is the identity map of an open set U ⊂ M. d Then, on U we have the vector field V (x) = ds s=0φs(x), which is called the vector field tangent to φs at s = 0. Let us recall that a vector field V on an open set U ⊂ M is called a Γ-vector field if its flow consists of elements of Γ.

Definition 7. If any vector field tangent to a curve in Γ is a Γ-vector field, then the pseudogroup is said to have the Lie pseudoalgebra [5]. If a pseudogroup Γ has Lie pseudoalgebra, then the set XΓ(U) of Γ-vector fields on U is a Lie subalgebra in the Lie algebra

X(U) of vector ﬁelds on U, and the correspondence U → XΓ(U) determines a sheaf XΓ of Lie algebras on M. 8 M.A.MALAKHALTSEV

Remark 2.3. Let Γ be the pseudogroup associated with an integrable first order structure s : M → L(M)/G. Then Γ has the Lie pseudoalgebra, and this pseudoalgebra coincides d ∗ with XΓ. The reason is that in the definition of the Lie derivative: LX s = dt |t=0φt (s) one can take any smooth curve of diffeomorphisms tangent to X.

A deformation of a Γ0-atlas A = {(Uα, φα)} is a family of Γ0-atlases A(s) = {(Uα, φα(s))}, where |s| < ε, such that n 1) Uα × (−ε, ε) → R , (x, s) → φα(s)(x) is smooth for each α; −1 2) φβα(s) = φβ(s)φα (s) is a curve in Γ0 for every α, β; 3) A(0) = A. −1 A deformation A(s) of a Γ0-atlas A is said to be nonessential if φα(s)φα lie in Γ0, i. e.

for every s, A(s) is compatible with the maximal Γ0-atlas determined by A.

Let A(s) = {(Uα, φα(s))} be a deformation of a Γ0-atlas A = {(Uα, φα)}. Then γβα(s) = −1 φβ φβα(s)φα is a curve in Γ and γβα(0) = Id. d ds s=0φβαφαβ(s) is a Γ0-vector field on φβα(φα(Uα ∩ Uβ)) ⊂ φβ(Uβ). W e define the vector field Vβα on Uα ∩ Uβ,

−1 d (11) Vβα(p) = dφβ φβα(s)(φα(p)) , ds s=0 and the vector ﬁeld Wα on Uα,

−1 d (12) Wα(p) = dφα φα(s)(p) . ds s=0

Statement 1. 1) V = W | − W | ; βα β Uα∩Uβ α Uα∩Uβ 2) {Vβα} is a 1-cocycle on the covering U = {Uα} with coeﬃcients in XΓ; 1 3) if the deformation A(s) is nonessential, the cohomology class [Vβα] ∈ H (M; XΓ) is trivial.

The cocycle {Vβα} is called an inﬁnitesimal deformation of the Γ0-structure. The class 1 [Vβα] ∈ H (M; XΓ) is called an essential inﬁnitesimal deformation of the Γ0-structure.

In general, not every inﬁnitesimal deformation is generated by a deformation of Γ0- structure. In [9] it is proved that if an essential inﬁnitesimal deformation represented by a cocycle {Vβα} is generated by a deformation of Γ0-structure, then the cocycle {Wαβγ =

[Vαβ, Vβγ]} of Γ-vector fields, where [ , ] is the Lie bracket of vector fields, represents the 2 2 trivial cohomology class in H (M; XΓ). The cohomology class [Wαβγ ] ∈ H (M, XΓ) is called the obstruction to integrability of the infinitesimal deformation {Vβα}. LECTURES ON DEFORMATIONS OF G-STRUCTURES 9

3. Fine resolution for the sheaf of Γ-vector fields.

We know that the space of infinitesimal essential deformations of a Γ-structure is 1 H (M; XΓ), which is defined in terms of the Cecˇ h complex. However, for calculations and for establishing relations with other geometrical structures we often need to have a ∗ differential complex whose cohomology is isomorphic to H (M; XΓ). Using the Spencer P -complex, we will construct such a complex for integrable first order G-structures.

3.1. Lie derivative of a section of a natural bundle. Let M be a manifold and E : Man → Bundles be a natural bundle. Let us take a section s : M → E(M) and

a vector field X ∈ X(M). Denote by φt the flow of X, and since the bundle E(M) is c natural, we have the one-parameter group φt : E(M) → E(M), and hence the complete lift Xc ∈ X(E(M)). Now denote by V E(M) → E(M) the vertical bundle of E(M), then the pullback bundle s∗(V E(M)) is a vector bundle over M, and for each p ∈ M there is ∗ defined the isomorphism Πp : (V E(M))s(p) → (s (V E(M))p. Then

def c (LX s)p = Π(X (s(p)) − dsp(X(p))),

is the Lie derivative of s with respect to X at a point p ∈ M.

From the definition it immediately follows that LX s = 0 if and only if X is an infini- c tesimal automorphism of s, i.e., if the flow φt preserves s.

Now let s: M → EG(M) be an integrable first order structure. We consider the Lie ∗ derivative LX s as a first-order differential operator DL : T M → s (V E). In [1], for a differential operator D : Γ(ξ) → ξ0, where ξ, ξ0 are vector bundles and Γ(ξ) is the sheaf of sections of ξ, a differential complex was constructed (the Spencer P-complex)

D 0 → Θ → Γ(ξ) −→ F 0 → F 1 → . . . ,

where Θ is the kernel of D. This construction, applied to a Lie derivative associated with a general pseudogroup structure, gives the deformation complex of this structure [1]. Then, specializing this complex to the case of the pseudogroup associated to an integrable

G-structure, we obtain the ﬁne resolution for the sheaf XΓ (for details, see [29]). In the next subsection we present this complex in terms of tensor ﬁelds and covariant derivatives.

3.2. P -complex for the Lie derivative. Let s : M → EG(M) = L(M)/G be an 1 integrable ﬁrst order G-structure. Let us denote by Eg the subbundle of the bundle T 1(M) consisting of linear operators whose matrices written with respect to the corresponding 1 Γ0-atlas lie in the Lie algebra g of G, and let Fg = T 1(M)/Eg. 10 M.A.MALAKHALTSEV

The P -complex of the Lie derivative is isomorphic to the complex (C q(P ), d), where k q Ω (M) ⊗ T M C (P ) = k−1 , Alt(Ω (M) ⊗ Eg) where Alt(tj ) = tj and the differential d : Cq(P ) → Cq+1(P ) is induced by i1...tk−1tk [i1...tk−1tk] the differential operator D = Alt ◦ ∇, where Alt is the alternation and ∇ is the covariant derivative of a torsion-free connection adapted to the G-structure s: ∇s = 0. This means that, with respect to local coordinates adapted to s (the charts of the Γ0-atlas), we have (Dω) j = ∇ ω j . i1...iq+1 [i1 i2...iq+1] The definition of d does not depend on an adapted connection choice because, if ∇(s) = k k k ∗ k−1 ∇(s) = 0, we have Tij = Γij −Γij in Eg⊗T M, so Alt(∇ω)−Alt(∇ω) lies in Alt(Ω (M)⊗

Eg).

Example 3.1 (C). For a complex structure, the complex (C q, d) is the Dolbeaut complex of vector-valued forms (see, e.g., [25]).

Example 3.2 (F). Let a foliation structure on a smooth manifold M be given by the 1 p integrable distribution ∆. Then Eg = {A ∈ T 1(M) | A(∆) ⊂ ∆}. Therefore C = Ωp(∆) ⊗ (T M/∆). If (xi, xα) are adapted local coordinates, i.e., if ∆ is given by the equations dxi = 0, then d can be written locally as (dω) i = ∂ ω j . Thus we α1...αq+1 [α1 α2...αq+1] arrive at Vaisman’s foliated cohomology [2].

Example 3.3 (S). Let us consider a symplectic manifold (M, θ). Then the subbundle Eg consists of those linear operators that are skew-symmetric with respect to θ: θ(AX, Y ) + 1 2 θ(X, AY ) = 0, and using the isomorphism T 1(M) → T (M) determined by θ, we obtain that Cq ∼= Ωq+1(M) and . The adapted connection ∇ satisfies ∇ω = 0 (a symplectic connection), and the differential d: Cq → Cq+1 is the exterior differential. Thus we find that the kernel of d is the Lie algebra of (locally) Hamiltonian vector fields.

∗ 3.3. Fine resolution for XΓ. It is clear that the construction of (C , d) transforms to the sheaf level. So we have the sequence of sheaves L i (s) 0 d 1 d 2 d (13) 0 → XΓ →− X −−→ C →− C →− C →− . . .

From the construction, it follows that these sheaves are ﬁne, since these ones are sheaves of modules over the ﬁne sheaf C∞. So we arrive at

Theorem 4. If the sheaf sequence (13) is exact (the “Poincare lemma” holds), then the 1 1 ∗ space of essential deformations H (M; XΓ) is isomorphic to H (C (M), d). LECTURES ON DEFORMATIONS OF G-STRUCTURES 11

4. Deformation of symplectic structure with Martinet singularities

4.1. Symplectic structure with singularities. Let ω0 be a closed diﬀerential 2-form 2n 2n on R such that Σ0 = {x ∈ R | det ω0(x) = 0} is an embedded manifold, and Γ0 be the 2n ∗ pseudogroup consisting of local diﬀeomorphisms f of R such that f ω0 = ω0.

A symplectic structure with singularities of type ω0 on a 2n-dimensional manifold M can be deﬁned in two equivalent ways.

Deﬁnition 8. A symplectic structure with singularities of type ω0 on a 2n-dimensional manifold M is a Γ0-structure on M, i. e. a maximal atlas whose transition functions lie in Γ0.

Deﬁnition 9. A 2-form ω on a 2n-dimensional smooth manifold M is called a symplectic structure with singularities of type ω0 if for any point p ∈ M there exist a neighborhood 2n 2n U(p) and a diﬀeomorphism φp : U(p) → O ⊂ R , where O is an open subset in R , such ∗ that ω|U(p) = φp(ω0).

The singular submanifold of a symplectic structure with singularities. Let a 2-form ω be a symplectic structure with singularities of type ω0 on a 2n-dimensional manifold M.

Lemma 1. The set Σ = {p ∈ M | det ω(p) = 0} is an embedded closed submanifold in

M, and dim Σ = dim Σ0.

The submanifold Σ will be called the singular submanifold of a symplectic structure with singularities.

Lemma 2. Let ω be a symplectic structure with singularities of type ω0 on a manifold M, and U ⊂ M be an open set. For V, W ∈ X(U), from iV ω = iW ω it follows that V = W .

4.2. Infinitesimal deformations of symplectic structure with singularities. We have defined the symplectic structure with singularities to be a pseudogroup structure (Definition 8), and in the equivalent way, to be a closed 2-form (Definition 9). Therefore, there are defined two spaces of essential infinitesimal deformations of symplectic structure with singularities. The first one is the space of essential infinitesimal deformations of pseu- 1 dogroup structure, which is isomorphic to the Cecˇ h cohomology group Hˇ (M; XΓ) with coefficients in the sheaf of Γ-vector fields, where Γ is the pseudogroup of automorphims of

symplectic structures with singularities. The second one is the space Dess(ω) of essential inﬁnitesimal deformations of the closed 2-form ω. In this section, for a compact manifold 1 M we construct a homomorphism φ : Hˇ (M; XΓ) → Dess(ω) and study its properties. In what follows we assume that M is compact. 12 M.A.MALAKHALTSEV

4.2.1. Infinitesimal deformations of a closed differential form. In 2.3 we have defined the deformations of pseudogroup structure and now we define the deformations of a closed differential form. Let η be a closed differential form on a manifold M. A deformation of η is a smooth family of closed forms η(s), |s| < ε, where η(0) = η. Let φ(s) be a flow of local diffeomorphisms of M. Then the family of forms η(s) = φ(s)∗(η) is called a nonessential deformation of η. A closed form θ is called a (nonessential) infinitesimal deformation of d η, if there exists a (nonessential) deformation η(s) of η such that θ = ds s=0η(s). Note that a closed form θ is a nonessential infinitesimal deformation of η if and only if there exists a vector field V ∈ X(M) such that θ = LV η = d(iV η). For any closed p-form θ, the family η(s) = η + sθ is a family of closed forms, and d p θ = ds s=0η(s), therefore the set of infinitesimal deformations D(η) of η ∈ Ω (M) coincides with the space of closed p-forms. The set of nonessential infinitesimal deformations is the subspace D0(η) ⊂ D(η) consisting of exact p-forms d(iV η), where V ∈ X(M). Then the quotient space Dess(η) = D(η)/D0(η) is called the space of essential infinitesimal deformations of the closed form η. From the definition of the space of essential infinitesimal deformations it follows that p for any closed p-form η we have the surjective homomorphism ψ : Dess(η) → H (M) which maps each essential infinitesimal deformation of η to the corresponding de Rham cohomology class.

4.3. Infinitesimal deformations of symplectic structure with singularities. Let us establish relationship between infinitesimal deformations of symplectic structure with singularities of type ω0 considered as a pseudogroup structure (see 2.3) and infinitesimal deformations of the corresponding closed 2-form ω (see 4.2.1).

Let a symplectic structure with singularities of type ω0 be given on a manifold M by a 2 Γ0-atlas A = {(Uα, φα)}. Let ω ∈ Ω (M) be the corresponding closed 2-form (see Defini- tions 8 and 9). Let us denote by Γ the pseudogroup consisting of local diffeomorphisms f of M such that f ∗ω = ω. Hence follows that a vector field V is a section of the sheaf

XΓ of Γ-vector ﬁelds if and only if LV ω = 0.

Each deformation A(s) = {(Uα, φα(s))} of the Γ0-atlas determines a deformation ω(s) of ∗ the form ω. Namely, the family of closed 2-forms ω(s) is given by ω(s)|Uα = φα(s)ω0. Let d η = ds s=0ω(s) be the corresponding infinitesimal deformation of ω, then η|Uα = LWα ω, where the vector field Wα ∈ X(Uα) is defined by (12). And Vαβ = Wα − Wβ is an infinitesimal deformation of A determined by the deformation of A(s) (see Statement 1). 1 Now we define a map φ from the space Hˇ (M, XΓ) of essential infinitesimal deformations of the Γ0-atlas to the space Dess(ω) of essential infinitesimal deformations of the closed LECTURES ON DEFORMATIONS OF G-STRUCTURES 13

2-form determined by this atlas. We proceed as follows. Let an essential inﬁnitesimal deformation be given by a cocycle of Γ-vector ﬁelds Vβα on the covering U = {Uα} of M.

Since the sheaf X of vector fields on M is fine, there exist vector fields Wα ∈ X(Uα) such that V = W − W . We set η = L ω, then, since L ω = 0, we get η | = η | . βα β α α Wα Vβα α Uαβ β Uαβ

Therefore, we get the form η such that η|Uα = ηα, and dη = 0. 0 0 0 0 If Vβα = Wβ − Wα, then on Uα ∩ Uβ we have Wβ − Wα = Wβ − Wα. Therefore, we 0 obtain the vector ﬁeld W such that W |Uα = Wα − Wα, and 0 η − η = LW ω = d(iW ω).

0 Thus, η and η determine the same class in the space Dess(ω). Thus we obtain the map

1 φ : Zˇ (U; XΓ) → Dess(ω), φ(Vβα) = [η].

0 From our construction folloe ws that, if cocycles Vβα ande Vβα are cohomologous, they deﬁne 0 the same closed form η, and, therefore, φ(Vβα) = φ(Vβα), i. e. we obtain the linear map ˇ 1 (14) φ : H (Me; XΓ) → De ess(ω).

Statement 2. Let us consider a symplectic structure ω with singularities of type ω0 given

by a Γ0-atlas A. Given a deformation A(s) of A, we denote by ω(s) the corresponding de- 1 formation of ω. If [Vβα] ∈ H (M, XΓ) is an essential inﬁnitesimal deformation determined d by A(s), and η = ds s=0ω(s), then φ([Vβα]) = [η].

Properties of the map φ.

Statement 3. φ is a monomorphism.

Let us deﬁne by Ωp the sheaf of p-forms on M. Let ι : X → Ω1 be the sheaf morphism 1 given by ιU : X(U) → Ω (U), ι(V ) = iV ω.

Statement 4. φ is surjective if and only if Ω1 = ι(X) + dΩ0.

4.4. The sheaf of local Hamiltonians and the map φ. The sheaf of local Hamilto- nians is a subsheaf T of the sheaf Ω0 of smooth functions on M. For any open set U a smooth function f lies in T (U) if and only if for any point p ∈ U there exist a neighbor- 0 0 hood U ⊂ U of p and V ∈ X(U ) such that df = iV ω. We will denote by i the inclusion

RM → T .

Let us construct a sheaf morphism π : T → XΓ. For f ∈ T (U) there exists a covering

U = {Uα} of U such that on each Uα there exists a vector ﬁeld Vα with the property

that df = iVα ω, which is uniquely deﬁned by virtue of Lemma 2. Then, on Uα ∩ Uβ we 14 M.A.MALAKHALTSEV

have Vα = Vβ. Thus, on U we obtain the vector ﬁeld V such that df = iV ω, which is

uniquely deﬁned. Furthermore, LV ω = d(iV ω) = d(df) = 0, therefore V ∈ XΓ(U). Thus

we have obtained the map πU : T (U) → XΓ(U), which is evidently linear. All the maps

πU determine a sheaf morphism π : T → XΓ. In fact T is the sheaf of algebras. The Poisson bracket { , } of sections of T is deﬁned in the following way. Let f, g ∈ T (U), df = iV ω, and dg = iW ω, where V, W ∈ XΓ(U) are uniquely deﬁned. Set

(15) {f, g} = dg(V ) − df(W ) = 2ω(W, V ).

Statement 5. i π 0 → RM −→ T −→ XΓ → 0 is an exact sequence of sheaves.

Corollary 1. 1. The sequence

0 δ 1 i∗ 1 π∗ (16) · · · → Hˇ (M; XΓ) → Hˇ (M; RM ) → Hˇ (M; T ) →

π∗ 1 δ 2 i∗ → Hˇ (M; XΓ) → Hˇ (M; RM ) → . . . is exact. 2) The diagram

1 φ Hˇ (M; XΓ) −→ Dess(ω) (17) δ ψ ˇ 2  R η 2  H (M; M ) −∼→ H (M), y = y is commutative. Here ψ is deﬁned in 4.2.1, δ is the connecting homomorphism of exact sequence (16), and η is the standard isomorphism between the Ceˇ ch cohomology with coeﬃcients in RM and the de Rham cohomology.

4.4.1. Obstructions to integrability of inﬁnitesimal deformations.

1 Statement 6. Let [v = {Vβα}] ∈ H (M; XΓ) be an essential inﬁnitesimal deformation of 2 a symplectic structure with singularities of type ω0, and [w = {Wαβγ}] ∈ H (M, XΓ) be the obstruction to integrability of this inﬁnitesimal deformation (see 1.1). Then [w] lies in 2 2 the image of the map π∗ : H (M; T ) → H (M; XΓ) (see (16)).

1 Let a 1-cocycle v ∈ Z (U; XΓ), where U is a covering of M consisting of contractible open sets, represent an inﬁnitesimal deformation of a symplectic structure with singularities 2 of type ω0. Then the 2-cocycle w = {Wαβγ = [Vαβ, Vβγ]} ∈ Z (U; XΓ) represents the LECTURES ON DEFORMATIONS OF G-STRUCTURES 15

1 obstruction to integrability of v. Given v, we have the cochain f = {fβα} ∈ C (U; T ) 2 such that π(f) = v (see Statement 5). Take the cochain gαβγ = {fαβ, fβγ } ∈ C (U; T ), where { , } is the Poisson bracket (15). Since π is a morphism between sheaves of Lie algebras, we obtain that π(g) = w.

4.5. Inﬁnitesimal deformations of symplectic structure with Martinet singularities.

4.5.1. Martinet singularities. In [16]–[19] it was proved that on R4 ﬁve generic types of germs of closed 2-forms exist, among them the following four types are stable: Type 0 1 2 3 4 ω0 = dx ∧ dx + dx ∧ dx , Type I 1 1 2 3 4 ω0 = x dx ∧ dx + dx ∧ dx , Type II-e (the elliptic type)

1 2 3 1 4 3 2 3 4 2 4 1 3 2 3 4 (18) ω0 = dx ∧ dx + x dx ∧ dx + x dx ∧ dx + x dx ∧ dx + (x − (x ) )dx ∧ dx ,

Type II-h (the hyperbolic type)

1 2 3 1 4 3 2 3 4 2 4 1 3 2 3 4 (19) ω0 = dx ∧ dx + x dx ∧ dx + x dx ∧ dx − x dx ∧ dx + (x − (x ) )dx ∧ dx .

One can characterize these types in the following way (see [16] for details). Let ω be a closed 2-form on a four-dimensional manifold M. Let us denote by Σ the set of points, where ω is degenerate. In what follows we assume that Σ is a three-dimensional submanifold of M. If a point p does not lie in Σ, then, by the Darboux theorem, ω has type 0 at p.

Now let p ∈ Σ and ω(p) =6 0. If the kernel Ep of ω(p) is transversal to Σ, then ω has type I at p. The set V = U ∩ Σ is open in Σ and consists of points of type I, and U \ V is everywhere dense in U and consists of points of type 0. 0 0 Let Σ ⊂ Σ be the set of points p such that Ep ⊂ TpΣ. If Σ is a submanifold in a 0 neighborhood of p, and Ep is transversal to Σ , then ω has type II-e, or II-h, at p. Let V = U ∩ Σ, W = U ∩ Σ0. Then U \ Σ is everywhere dense in U and consists of points of type 0, V \ W is everywhere dense in V and consists of points of type I, and W consists of points of type II. Let us consider a closed 2-form ω on a four-dimensional manifold M. If all points of M has type 0, then (M, ω) is a symplectic manifold. If any point of M has type 0 or I, then (M, ω) will be called a symplectic manifold with Martinet singularities of type I. If 16 M.A.MALAKHALTSEV

any point of M has type 0, I, or II, then (M, ω) will be called a symplectic manifold with Martinet singularities of type II. Note that a symplectic manifold (M, ω) with Martinet singularities is a symplectic

manifold with singularities of type ω0, where ω0 is given by (18) or (19). Indeed, let 1 2 3 4 1 φ(p) = (x0, x0, x0, x0). Then p has type 0 if and only if x0 =6 0; p has type I if and only 1 3 2 4 2 1 3 if x0 = 0 and (x0) + (x0) =6 0; and p has type II if and only if x0 = 0, x0 = 0, and 4 x0 = 0. Thus the symplectic structure is a partial case of the symplectic structure with Martinet singularities of type I, which in turn is a partial case of the symplectic structure with Martinet singularities of type II. Example of compact manifold with symplectic structure having Martinet singularity. Let us take the four-dimensional torus T4, and let π : R4 → T4 be the standard covering. Let us denote by θa, a = 1, 4, the 1-forms on T4 such that π∗(θa) = dxa. Set ω = fθ1 ∧ θ2 + θ3 ∧ θ4, where π∗f = cos(x1). It is clear that dω = 0. Let us denote by 1 3 3 p1, p2 ∈ S the zeroes of f. Then Σ = p1 × T ∪ p2 × T , and the restriction of ω to each connected component of Σ is θ3 ∧ θ4, i. e. it has rank 2. Then ω determines a symplectic structure with Martinet singularity on T4. Type 0 If ω is a symplectic structure without singularities, then the sheaf T coincides with the sheaf of smooth functions Ω0, therefore Hk(M; T ) = 0 for k > 0. Furthermore, ι : X → Ω1 is a sheaf isomorphism (see 1.4). Hence, Corollary 1 and Statement 4 imply 1 ∼ ∼ 2 the well-known fact that for a symplectic structure H (M; XΓ) = Dess(ω) = H (M). Type I

1 Statement 7. For a symplectic structure ω with Martinet singularities, φ : H (M; XΓ) →

Dess(ω) is an isomorphism.

Type II In this case the homomorphism φ is not surjective. Let ω be a symplectic structure with singularities of type

1 2 3 1 4 3 2 3 4 2 4 1 3 2 3 4 ω0 = dx ∧ dx + x dx ∧ dx + x dx ∧ dx + x dx ∧ dx + (x − (x ) )dx ∧ dx

3 2 4 Then the form α = (x ) dx cannot be represented as iV ω + df, therefore, by Statement 4, φ is not surjective.

4.5.2. Calculation of H ∗(M; T ) for Martinet singularities. On the submanifold Σ of singular points we have the foliation F with singularities whose regular leaves are integral curves of the one-dimensional distribution on Σ \ Σ0 obtained by the intersection of the LECTURES ON DEFORMATIONS OF G-STRUCTURES 17 kernel of ω with the spaces tangent to Σ, and whose singular leaf is Σ0. Let us denote by

Fb the sheaf of basic functions of the foliation F, i. e. the sheaf of functions which are locally constant along the leaves of F.

Let µ be a volume form on M. Then ω ∧ ω = Pfµ(ω)µ, where Pfµ(ω) is called the Pfaﬃan of ω with respect to µ. It is evident that, if µ0 = λµ is another volume form, ∞ where λ is nonvanishing function, then Pf µ(ω) = λPfµ0 (ω). Hence, in the ring sheaf C of smooth functions we have the subsheaf I of principal ideals generated by Pf µ(ω):

∞ I(U) = {Pfµ(ω)|U · f | f ∈ C (U)}, which does not depend on the choice of µ. Note that I is the ideal sheaf whose sections are the functions vanishing on Σ, i. e. C∞/I is the sheaf of smooth functions on Σ. Let X be a topological space, A ⊂ X be a closed subset, and G be a ring sheaf on A. Then we have the sheaf GX on X generated by the presheaf: U → 0 if U ∩ A = ∅, otherwise U → G(U ∩ A) (see [24]). ∞ ∞ M Let r : CM → (CΣ ) be the sheaf morphism determined by the restriction of functions ∞ M ∞ ∞ M to Σ. This means that, if U ∩Σ = ∅, then (CΣ ) (U) = 0 and rU : CM (U) → (CΣ ) (U) is 0 ∞ ∞ 0 the zero homomorphism; if U ∩ Σ = U =6 ∅, then rU : CM (U) → CΣ (U ) is the restriction ∞ 0 2 ∞ M of f ∈ C (U) to U . Now, let us denote by i the inclusion I → CM , and set Fb = Fb .

Statement 8. e

2 i r (20) 0 → I →− T →− Fb → 0. is the exact sequence of sheaves on M. e

Corollary 2. For a symplectic manifold (M, ω) with Martinet singularities,

q q H (M; T ) ∼= H (Σ; Fb), q > 0, where T is the sheaf of local Hamiltonians on M, and Fb is the sheaf of basic functions of the foliation with singularities induced by the kernel of ω on the singular submanifold Σ.

Corollary 3. For a symplectic manifold (M, ω) with Martinet singularities such that Σ0 = ∅, q ∼ q+1 H (M; Xh) = HDR (M), q > 2, k where Xh is the sheaf of Hamiltonian vector ﬁelds on M, HDR(M) is the de Rham cohomology of M. 18 M.A.MALAKHALTSEV

q ∼ q+1 Corollary 4. If the foliation F is a ﬁber bundle, then H (M; XΓ) = HDR (M) for q > 0. 2 ∼ In particular, if HDR(M) = 0, then ω is inﬁnitesimally rigid.

Example 4.1. Let (M , ω ) be a symplectic 2n-dimensional manifold, M be a 2n-dimensional manifold, and π : M → M be a smooth map. Let ω = π∗ω. Then ω is a closed 2-form c on M, and the followingb diagram c α T M −−−ω→ T ∗M

dπ dπ∗

 αωˆ ∗x TM −−−→ T M . is commutative. Then y  c c −1 −1 ∗ (21) Ann(ω) = ker αω = dπ (αωˆ ker dπ ). Now let M be R4 (with coordinates yα, α = 1, 4) endowed with the standard symplectic structure c ω = dy1 ∧ dy2 + dy3 ∧ dy4, 5 M = S4 ⊂ R5 be the standard sphere given by (xi)2 = 1, and π : S4 → R4 be the b i=1 restriction of the projection R5 → R4, yα = xα, α P= 1, 4. R5 5 S3 S4 Let Π be the hyperplane in given by x = 0, and 0 = ∩ Π be the equator of S4 S4 R4 S4 S3 . It is clear that dπp : Tp → Tp is an isomorphism for any p ∈ \ 0, hence ω is S4 S3 nondegenerate on \ 0. Further, by a coordinate calculation with the use of (21), we S3 2 1 4 3 get that at points of 0 the kernel of ω is spanned by ∂5 and −x ∂1 + x ∂2 − x ∂3 + x ∂4. t S3 Hence Ann(ω) 0, and the one-dimensional foliation F corresponding to the distribution S3 S3 ker(ω) ∩ T 0 on 0 is the Hopf bundle. Then ω is a symplectic structure with Martinet singularities of type I, which satisﬁes the assumptions of Corollary 4. Hence ω is rigid.

4.5.3. Diﬀerential complex associated to symplectic form with Martinet singularities. Here we expose results of [30]. We start with the following standard algebraic construction. Let K be a ring and d : K → K be a diﬀerentiation such that d2 = 0. Let I be an ideal in K, then

0 I = {a + ki dbi | a, bi ∈ I}

also is an ideal in K such that d : I 0 → I0. Then we have the following exact sequence of diﬀerential rings: 0 → I0 → K → K/I0 → 0 The same construction can be done for sheaves of rings over a smooth manifold M. Then, for a sheaf of rings K over M endowed with a diﬀerential d such that d2 = 0, and LECTURES ON DEFORMATIONS OF G-STRUCTURES 19 a subsheaf ι : I ,→ K of ideals, we get the sheaf I0 of ideals and the exact sequence of sheaves 0 → I0 → K → K/I0 → 0

and the corresponding cohomology exact sequence

· · · → Hk(M; I0) → Hk(M; K) → Hk(M; K/I0) → Hk+1(M; I0) → . . .

For any vector bundles ξ : Eξ → M and η : Eη → M, each vector bundle morphism

Q : ξ → η determines the morphism Q : Γξ → Γη of sheaves of vector spaces: for each

s ∈ Γξ(U), the section Q(s)(p) = Qp(s(p)), p ∈ U, lies in Γη(U). The kernel of Q is the ∞ sheaf K(U) = {s ∈ Γξ(U) | Q(s) = 0} of modules over the ﬁne sheaf C , therefore K is

also ﬁne. Now let us consider the presheaf F(U) = {t ∈ Γη(U) | t = Q(s)}.

Lemma 3. The presheaf F is a sheaf.

q Let ξ : Eξ → M be a vector bundle, and A : ξ → Λ M be a vector bundle morphism.

Denote by A : Γξ → ΩM the sheaf morphism corresponding to A. Then the subsheaf q A(Γξ) generates the subsheaf IM ⊂ ΩM of ideals. Let us denote by FM = ⊕FM ⊂ ΩM 0 the corresponding graded sheaf IM of diﬀerential ideals. We take an open U ⊂ M such that ξ and ΛM are trivial over U. Then, on U we get ∞ q-forms ωa, a = 1, rankξ, which span A(U) over the ring C (U) of functions on U. One can easily see that

k a a a k a k−1 (22) F (U) = {φ ∧ ωa + ψ ∧ dωa | φ ∈ Ω (U), ψ ∈ Ω (U)}

Thus, to any morphism A : ξ → ΛqM we associate the complex (F ∗, d) of sheaves,

which is a subcomplex of the de Rham complex (ΩM , d) considered also as a complex of sheaves. Also, we have the exact sequence of sheaves

0 → F → ΩM → G = ΩM /F → 0.

1 Remark 4.1. Let q = 1, and A : ξ → Ω be a morphism. Then the sheaf A(Γξ) is the sheaf of sections of a subbundle (with singularities) in T ∗M. If rankA is constant, then A determines a distribution on M, and if, in addition, this distribution is integrable, (F ∗(M), d) is the complex generated by the basic forms of the corresponding foliation, which is widely used in the foliation theory [26], [2].

Remark 4.2. If A : ξ → Ωq is surjective, then the associated complex (F ∗(M), d) is the de Rham complex of M. 20 M.A.MALAKHALTSEV

Hereafter we assume that A is surjective on an open everywhere dense set M \Σ, where ι : Σ ,→ M is an embedded submanifold. Then, for each open U such that U ∩ Σ = ∅ we have F k(U) = Ωk(U), hence G(U) = 0. From this follows that the sheaf G is supported on Σ. q+1 For a closed ω ∈ Ω (M), the Lie derivative D(V ) = LV ω = diV ω is a ﬁrst order diﬀerential operator T M → Λq+1M. The operator D can be included to the complex of q sheaves associated to the vector bundle morphism Iω : T M → Λ , I(V ) = iV ω:

D 1 d 2 d (23) 0 → XΓ → XM −→ F −→ F −→ . . . , where XΓ is the sheaf of inﬁnitesimal automorphisms of ω. Evidently, all the sheaves in

(23), except for XΓ are ﬁne. Therefore, if (23) is locally exact, it gives a ﬁne resolution

for XΓ. However, in general, (23) fails to be locally exact. Let ω be a closed 2-form on a smooth manifold M such that det ω = 0 on a closed submanifold i : Σ ,→ M and rankω = 2m is constant on Σ. Then ω determines the vector 1 bundle morphism Iω : T M → Λ M, Iω(V ) = iV ω, which is a vector bundle isomorphism

over M \ Σ. The kernel of Iω is a vector bundle over Σ, call it : E → Σ. Denote by 1 Iω the corresponding sheaf morphism XM → ΩM . From Lemma proved in [28] it follows

that τ ∈ Iω(XM ) if and only if τ|E = 0. We will consider the symplectic structures with Martinet singularities. Let ω be a closed 2-form on a 2n-dimensional manifold. Assume that for each point p ∈ M one can take a chart (U, ui) such that

(24) ω = u1du1 ∧ du2 + du3 ∧ du4 · · · + du2n−1 ∧ du2n.

Statement 9. Let (F ∗, d) be the complex of sheaves associated to the symplectic form ω 1 1 with Martinet singularities locally given by (24). Then F is the subsheaf of ΩM consisting k k of forms which vanish on the subbundle E ⊂ T M|Σ, and F = ΩM for k ≥ 2.

2 Statement 10. For D : XM → ΩM , D(V ) = LV ω, the sequence of sheaves (see (23))

i D 2 d 3 d (25) 0 → XΓ −→ XM −→ Ω −→ Ω −→ . . .

is a ﬁne resolution for the sheaf XΓ.

q ∼ Corollary 5. For ω with Martinet singularities locally given by (24), H (M; XΓ) = q+1 HDR (M), q ≥ 1, where HDR(M) is the de Rham cohomology.

Let us consider another type of Martinet singularities. Let ω be a closed 2-form on a four-dimensional manifold, and assume that for each point p ∈ M one can take a chart LECTURES ON DEFORMATIONS OF G-STRUCTURES 21

(U, ui) such that

1 2 3 1 4 ω0 =du ∧ du + u du ∧ du (26) + u3du2 ∧ du3 + u4du2 ∧ du4 + (u1 − (u3)2)du3 ∧ du4,

Statement 11. Let (F ∗, d) be the complex of sheaves associated to the symplectic form ω 1 1 with Martinet singularities locally given by (26). Then F is the subsheaf of ΩM consisting k k of forms vanishing on the subbundle E ⊂ T M|Σ, and F = ΩM for k ≥ 2.

Thus, in this case we also get the complex of sheaves (25). However, in this case the complex (25) is not locally exact.

References

[1] Pommaret, J.F. Systems of partial diﬀerential equations and Lie pseudogroups, Math. and Appl., 14 (1978).

[2] Vaisman I. df -cohomologies of Lagrangian foliations, Monatshefte fur¨ Math., 106 (1988), pp. 221– 244. [3] Martinet J. Sur les singularités des formes différetielles // Ann. Inst. Fourier. Grenoble. – 1970. – V. 20. – No 1. – P. 95–178. [4] Arnol’d, V.I. Singularities of caustics and wave fronts. (Osobennosti kaustik i volnovykh frontov). (Russian) Biblioteka Matematika. 1. Moskva: Fazis. x+ 334 p. (1996). [ISBN 5-7036-0021-9] [5] Spencer D.C. Deformations of structures on manifolds defined by transitive continuous pseudogroups. I. Infinitesimal deformations of structure // Ann. Math. – 1962. – V. 76. – No 2. – P. 306–398. [6] Spencer D.C. Deformations of structures on manifolds defined by transitive continuous pseudogroups. II. Deformations of structure // Ann. of Math. – 1962. – V. 76. – No 3. – P. 399–445. [7] Spencer D.C. Deformations of structures on manifolds defined by transitive continuous pseudogroups. III. Structures defined by elliptic pseudogroup // Ann. Math. – 1965. – V. 81. – No 2. – P. 389–450. [8] Guillemin V., Sternberg S. Deformation theory of pseudogroup structures // Mem. Am. Math. Soc. – 1968. – V. 64. – 80 p. [9] Kodaira K. Complex manifolds and deformations of complex structures. – Springer, 1986. – 400 p. [10] Fomenko, A.T. Symplectic geometry. Methods and applications. (Simplekticheskaya geometriya. Metody i prilozheniya). (Russian) Moskva: Izdatel’stvo Moskovskogo Universiteta. 414 p. (1988). [ISBN 5-211-00083-8] [11] Warner, Frank W. Foundations of differentiable manifolds and Lie groups. (1987). [12] , Wells R.O.jun. Differential analysis on complex manifolds. 2nd ed. (English) Graduate Texts in Mathematics. 65. New York - Heidelberg - Berlin: Springer-Verlag. X, 260 p. (1980). [13] Sternberg, Shlomo Lectures on differential geometry. 2nd ed. (English) New York: Chelsea Publishing Company. XV, 442 p. (1983). [14] Kobayashi, Shoshichi Transformation groups in differential geometry. Reprint of the 1972 ed. (Eng- lish) Classics in Mathematics. Berlin: Springer-Verlag. viii, 182 p. 22 M.A.MALAKHALTSEV

[15] Albert, C.; Molino, P. Pseudogroupes de Lie transitifs. II: Th’eor‘ems d’int’egrabilitè. (Transitive Lie pseudogroups. II: Integrability theorems). (English) Travaux en Cours, 19. Paris: Hermann. VIII, 131 p. [16] Martinet J. Sur les singularités des formes différentielles // Ann. Inst. Fourier (Grenoble). – 1970. – V. 20. – No 1. – P. 95–178. [17] Martinet J. Singularities of smooth functions and maps // London Math. Soc. Lecture Note Ser. –1982. – No 58. – 256 p. [18] Roussarie R. Modèles locaux de champs et de formes // Astérisque. – 1975. – No 30. – 181 p. [19] Golubitsky M., Tischler D. An example of moduli for singular symplectic forms // Invent. Math. – 1977. – V. 38. – No 3. – P. 219–225. [20] Cannas da Silva A., Guillemin V., Woodwart C. On the unfolding of folded symplectic structures // Math. Res. Lett. – 2000. – V. 7. – No 1. – P. 35–53. [21] Cannas da Silva A. Fold-forms for four-folds // Preprint. – 2003. – 16 p. [22] Domitrz W. Non-local invariants of Martinet’s singular symplectic structure // Banach Center Publ. – 2002. – V. 60. – P. 122–143. [23] Fomenko, A.T. Symplectic geometry. Methods and applications. (Simplekticheskaya geometriya. Metody i prilozheniya). (Russian) Moskva: Izdatel’stvo Moskovskogo Universiteta. 414 p. (1988). [ISBN 5-211-00083-8] [24] Bredon, G. E. Sheaf theory. McGraw-Hill New York 1967 [25] Griffiths, Phillip; Harris, Joseph Principles of algebraic geometry. 2nd ed. (English) Wiley Classics Library. New York, NY: John Wiley & Sons Ltd. xii, 813 p. [26] Molino, Pierre Riemannian foliations. Progress in Mathematics, Vol. 73. Boston-Basel: Birkhäuser. XII, 339 p. [27] Malakhaltsev, M.A. Infinitesimal deformations of a symplectic structure with singularities. Russ. Math. 47, No.11, 38-46 (2003); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2003, No.11, 42-50 (2003). [28] Malakhaltsev, M.A. Sheaf of local Hamiltonians of symplectic manifolds with Martinet singularities. Russ. Math. No.11, 45-52 (2004) [29] Malakhaltsev, M.A. The Lie derivative and cohomology of G-structures. Lobachevskii Journal of Mathematics, 1999, Vol.3, pp.215-220 (http://ljm.ksu.ru). [30] Malakhaltsev, M.A. Differential complex associated to closed differential forms of nonconstant rank Lobachevskii Journal of Mathematics, 1999, Vol.23, pp.183-192 (http://ljm.ksu.ru).

Kazan State University, Kazan, Russia; Universidad de Los Andes, Bogota, Colombia E-mail address: [email protected]